How to convert mouse input co-ordinates to the orthographic plane

OpenGL OpenGL: Basic Coding
1 
    case WM_MOUSEMOVE:
    {
        // Get the mouse position
        int mouseX = GET_X_LPARAM(lParam);
        int mouseY = GET_Y_LPARAM(lParam);

        // Convert mouse coordinates to NDC
        float NDC_X = (2.0f * static_cast<float>(mouseX)) / window_width - 1.0f;
        float NDC_Y = 1.0f - (2.0f * static_cast<float>(mouseY)) / window_height;

        // Convert NDC to orthographic coordinates
        glm::vec4 orthoCoords = glm::inverse(projection) * glm::vec4(NDC_X, NDC_Y, 0.0f, 1.0f);
        float glY = orthoCoords.x;
        float glX = orthoCoords.y;

        // Update the vertices of the two lines
        std::vector<Vertex> line1_vertices = {
            { static_cast<float>(-window_width) / 2.0f, glY, 0.0f, 1.0f, 1.0f, 0.0f, 0.0f, 1.0f, 0.0f, 0.0f },
            { static_cast<float>(window_width) / 2.0f, glY, 0.0f, 1.0f, 1.0f, 0.0f, 0.0f, 1.0f, 1.0f, 0.0f }
        };
        std::vector<Vertex> line2_vertices = {
            { glX, static_cast<float>(window_height) / 2.0f, 0.0f, 1.0f, 1.0f, 0.0f, 0.0f, 1.0f, 0.0f, 0.0f },
            { glX, static_cast<float>(-window_height) / 2.0f, 0.0f, 1.0f, 1.0f, 0.0f, 0.0f, 1.0f, 1.0f, 0.0f }
        };

        // Assuming the first two drawables are the two lines
        Line* line1 = dynamic_cast<Line*>(drawables[0].get());
        Line* line2 = dynamic_cast<Line*>(drawables[1].get());

        if (line1 && line2) {
            line1->updateVertices(line1_vertices);
            line2->updateVertices(line2_vertices);
        }

        break;
    }

How is it possible to convert the mouse co-ordinates from this win32 API example to the orthographic plane?

So far the orthographic projection I have set up has the center of the client area as x=0,y=0 and left is -window_width/2 and right is window_width/2 and top is window_height/2 and bottom is -window_height/2.

So far what I have here is not working yet. Is there a known formula or method to perform such a conversion?

2 
        int mouseX = GET_X_LPARAM(lParam);
        int mouseY = GET_Y_LPARAM(lParam);

        float NDC_X = 2.0f * static_cast<float>(mouseX) / static_cast<float>(window_width) - 1.0f;
        float NDC_Y = 1.0f - 2.0f * static_cast<float>(mouseY) / static_cast<float>(window_height);

        glm::vec4 orthoCoords = glm::inverse(projection) * glm::vec4(NDC_X, NDC_Y, 0.0f, 1.0f);
        float glX = orthoCoords.x;
        float glY = orthoCoords.y;

This does seem to work so far.