Setting quarkus.resteasy.path not reflected in OpenAPI or Swagger
### Describe the bug

When I set the "quarkus.resteast.path" it works making API calls directly but does not get reflected in the OpenAPI or Swagger.

Using @ApplicationPath and setting eash @Path looks to work.

### Expected behavior

OpenAPI and Swagger reflect the path set in the application properties.

### Actual behavior

RESTful calls works but OpenAPI and Swagger do not reflect the path.  From Swagger invoking an endpoint gives a 404.

### How to Reproduce?

_No response_

### Output of `uname -a` or `ver`

_No response_

### Output of `java -version`

_No response_

### GraalVM version (if different from Java)

_No response_

### Quarkus version or git rev

_No response_

### Build tool (ie. output of `mvnw --version` or `gradlew --version`)

_No response_

### Additional information

_No response_
diff --git a/extensions/smallrye-openapi/deployment/src/main/java/io/quarkus/smallrye/openapi/deployment/SmallRyeOpenApiProcessor.java b/extensions/smallrye-openapi/deployment/src/main/java/io/quarkus/smallrye/openapi/deployment/SmallRyeOpenApiProcessor.java
index 2a96133a559..276e9e3619c 100644
--- a/extensions/smallrye-openapi/deployment/src/main/java/io/quarkus/smallrye/openapi/deployment/SmallRyeOpenApiProcessor.java
+++ b/extensions/smallrye-openapi/deployment/src/main/java/io/quarkus/smallrye/openapi/deployment/SmallRyeOpenApiProcessor.java
@@ -496,17 +496,22 @@ private OpenAPI generateAnnotationModel(IndexView indexView, Capabilities capabi
         OpenApiConfig openApiConfig = new OpenApiConfigImpl(config);
 
         List<AnnotationScannerExtension> extensions = new ArrayList<>();
+
         // Add the RESTEasy extension if the capability is present
+        String defaultPath = httpRootPathBuildItem.getRootPath();
         if (capabilities.isPresent(Capability.RESTEASY)) {
             extensions.add(new RESTEasyExtension(indexView));
+            if (resteasyJaxrsConfig.isPresent()) {
+                defaultPath = resteasyJaxrsConfig.get().getRootPath();
+            }
+        } else if (capabilities.isPresent(Capability.RESTEASY_REACTIVE)) {
+            extensions.add(new RESTEasyExtension(indexView));
+            Optional<String> maybePath = config.getOptionalValue("quarkus.resteasy-reactive.path", String.class);
+            if (maybePath.isPresent()) {
+                defaultPath = maybePath.get();
+            }
         }
 
-        String defaultPath;
-        if (resteasyJaxrsConfig.isPresent()) {
-            defaultPath = resteasyJaxrsConfig.get().getRootPath();
-        } else {
-            defaultPath = httpRootPathBuildItem.getRootPath();
-        }
         if (defaultPath != null && !"/".equals(defaultPath)) {
             extensions.add(new CustomPathExtension(defaultPath));
         }
['extensions/smallrye-openapi/deployment/src/main/java/io/quarkus/smallrye/openapi/deployment/SmallRyeOpenApiProcessor.java']