WebFlux with WebFilter - correctly starting a long running task inside of a WebFilter without blocking response

Given a org.springframework.web.server.WebFilter - what are some best ways to start a "longer running" thread without blocking the REST response coming back to the consumer. The long running process in my case will be an IO operation persisting some data in a database. It wont really take "very long" but I do not want to impact the response time.

Ive tried two different approaches and they seem to both work, not sure which one is considered "correct".

  // approach #1
  @Override
  @NonNull
  public Mono<Void> filter(ServerWebExchange exchange, WebFilterChain chain) {

    String requestId = UUID.randomUUID().toString();
    String correlationId = UUID.randomUUID().toString();

    return chain.filter(exchange)
        .doAfterTerminate(()->{
              Mono.fromRunnable(new LongRunningTask(requestId, 10000L))
                  .subscribeOn(Schedulers.boundedElastic())
                  .subscribe();
        })
        .contextWrite(Context.of(
            KEY_REQUEST_ID, requestId,
            KEY_CORRELATION_ID, correlationId
        ));

  }

// approach #2

  @Override
  @NonNull
  public Mono<Void> filter(ServerWebExchange exchange, WebFilterChain chain) {

    String requestId = UUID.randomUUID().toString();
    String correlationId = UUID.randomUUID().toString();
    
    return chain.filter(exchange)
        .publishOn(Schedulers.boundedElastic())
        .then(Mono.fromRunnable(new LongRunningTask(requestId, 10000L)).then())
        .contextWrite(Context.of(
            KEY_REQUEST_ID, requestId,
            KEY_CORRELATION_ID, correlationId
        ));
    
  }