Send message from serverless application to client through WebSocket

There is an API Gateway with Regional endpoint and WebSocket protocol that integrated with a lambda function. I can open wss://[gatewayId].execute-api.[region].amazonaws.com/[stage]/ connection, send a message to it and get the ConnectionId in the response. The response is getting in (new WebSocket(wssUri)).addEventListener('message', (event) => ...) on the client; the same as direct messages from server should be got. To send a message to the connected client, the application sends a request to HTTPS endpoint in the AGI Gateway. The call should be signed, so AWSSDK.ApiGatewayManagementApi package is used (.NET6, C#):

•       var  connectionId = "...";
        var config = new AmazonApiGatewayManagementApiConfig
        {
            ServiceURL = "https://[gatewayId].execute-api.[region].amazonaws.com/[stage]",
            RegionEndpoint = Amazon.RegionEndpoint.GetBySystemName("[region]")
        };
        var client = new AmazonApiGatewayManagementApiClient(config);
        var postToConnectionRequest = new PostToConnectionRequest
        {
            ConnectionId = connectionId,
            Data = new MemoryStream(System.Text.Encoding.UTF8.GetBytes("Hello from C#!"))
        };
        await client.PostToConnectionAsync(postToConnectionRequest);
  

*The code is run either in the integrated with the API Gateway lambda function or other lambda function; they both have a role with "AmazonAPIGatewayInvokeFullAccess" permission. Any case, the PostToConnectionAsync call in the last string fails due to timeout (30s). As I see the same can be done from a local console application, but new BasicAWSCredentials(accessKey, secretKey) parameter should be added to AmazonApiGatewayManagementApiClient constructor to sign the call. Then I can wait for the exception: System.Net.Http.HttpRequestException: The requested name is valid, but no data of the requested type was found. (execute-api.[region].amazonaws.com:443). There are little info about the question and nothing in C#. Could you help, what is wrong?