404 errors GCP API gateway multiple cloud run backends

Question

I'm trying to secure multiple cloud run services with gcp Api gateway, which integrates firebase jwt auth. I attempted at first securing just one API with the following schema and all went fine:

    # openapi2-run.yaml
swagger: '2.0'
info:
  title: memsy-gateway
  description: Sample API on API Gateway with a Cloud Run backend
  version: 1.0.0
schemes:
  - https
consumes:
      - application/json
produces:
  - application/json
x-google-backend:
  address: https://mnemonic-api-staging-ue.a.run.app
securityDefinitions:
  jwt_auth:
    authorizationUrl: ''
    flow: 'implicit'
    type: 'oauth2'
    x-google-issuer: 'https://securetoken.google.com/the-journey-method'
    x-google-jwks_uri: 'https://www.googleapis.com/service_accounts/v1/metadata/x509/[email protected]'
    x-google-audiences: 'the-journey-method, https://mnemonic-api-staging-ue.a.run.app'
paths:
  /mnemonic-api:
    post:
      security:
        - jwt_auth: []
      summary: Mnemonic API
      operationId: mnemonics
      parameters:
      - in: body
        name: input
        description: string to process
        schema:
          $ref: '#/definitions/InputString'
      responses:
        '200':
          description: A successful response
          schema:
            type: object
        '400':
          description: invalid input, object invalid
    options:
      operationId: create-cors
      responses:
        '200':
          description: Success
            
definitions:
  InputString:
    type: object
    properties:
      input:         
        type: string
      title:         
        type: string
    required:
      - input
      - title

I then tried to secure two services using the following schema but am now getting 404 errors on the paths in the config. I can also access the backends via their cloud run urls without any jwt token, so I'm wondering what's wrong in my config?

  # openapi2-run.yaml
swagger: '2.0'
info:
  title: memsy-gateway
  description: Sample API on API Gateway with a Cloud Run backend
  version: 1.0.0
schemes:
  - https
consumes:
      - application/json
produces:
  - application/json
securityDefinitions:
  jwt_auth:
    authorizationUrl: ''
    flow: 'implicit'
    type: 'oauth2'
    x-google-issuer: 'https://securetoken.google.com/the-journey-method'
    x-google-jwks_uri: 'https://www.googleapis.com/service_accounts/v1/metadata/x509/[email protected]'
    x-google-audiences: 'the-journey-method, https://mnemonic-api-staging-ue.a.run.app, https://backend-dql-flask-uc.a.run.app'
paths:
  /mnemonic-api:
    post:
      security:
        - jwt_auth: []
      summary: Mnemonic API
      operationId: mnemonics
      x-google-backend:
        address: https://mnemonic-api-staging-ue.a.run.app
      parameters:
      - in: body
        name: input
        description: string to process
        schema:
          $ref: '#/definitions/InputString'
      responses:
        '200':
          description: A successful response
          schema:
            type: object
        '400':
          description: invalid input, object invalid
    options:
      operationId: create-cors
      responses:
        '200':
          description: Success
  /dql/deleteFolder:
    post:
      security:
        - jwt_auth: []
      summary: Dql
      operationId: deleteFolder
      x-google-backend:
        address: https://backend-dql-flask-uc.a.run.app
      parameters:
      - in: body
        name: input
        description: user and id strings
        schema:
          $ref: '#/definitions/Dql'
      responses:
        '200':
          description: A successful response
          schema:
            type: object
        '400':
          description: invalid input, object invalid
    options:
      operationId: create-cors-dql
      responses:
        '200':
          description: Success
            
definitions:
  InputString:
    type: object
    properties:
      input:         
        type: string
      title:         
        type: string
    required:
      - input
      - title
  Dql:
    type: object
    properties:
      user:         
        type: string
      id:         
        type: string
    required:
      - user
      - id

Answer 1

The documentation is not clear honestly - with trial and error I could make it work by matching the path on the openapi definition with the path in my webserver router instead of giving the openapi definition the host only, for example:

paths:
  /mnemonic-api:
    post:
      security:
        - jwt_auth: []
      summary: Mnemonic API
      operationId: mnemonics
      x-google-backend:
        address: https://mnemonic-api-staging-ue.a.run.app/mnemonic-api