OpenGL paint engine fails to render images bigger than the maximum texture size

#include <QApplication>
#include <QPaintEngine>
#include <QImage>
#include <QPainter>
#include <QWidget>

class Widget : public QWidget
{
    Q_OBJECT
public:
    Widget(QWidget* parent = 0);

protected:
    void paintEvent (QPaintEvent* event);

private:
    QImage m_image;
    int m_imgWidth;
    int m_imgHeight;
};

Widget::Widget(QWidget* parent)
    :QWidget(parent),
    m_imgWidth(2560), // fails to render if width or height > 2048
    m_imgHeight(1440) 
{
    QRect rect(0, 0, m_imgWidth, m_imgHeight);
    m_image = QImage(rect.size(), QImage::Format_RGB32);

    QPainter painter(&m_image);
    QColor color(200, 240, 160);
    QBrush brush(color);
    painter.setBrush(brush);
    painter.setPen(color);
    painter.drawRect(rect);

    // draw a checkerboard pattern on image
    color = QColor(240, 160, 200);
    brush.setColor(color);
    painter.setPen(color);
    painter.setBrush(brush);
    int length = 157;
    int y = 0;
    int c = 0;
      while(y*length+length < m_imgHeight) {
        int x = c % 2;        
        while(x*length+length < m_imgWidth) {
            rect.setCoords(x*length, y*length, x*length+length, y*length+length);
            painter.drawRect(rect);
            x +=2;
        }
        ++y;
        ++c;
    }
}

void Widget::paintEvent(QPaintEvent* event) {
    Q_UNUSED(event)
    QPainter painter(this);
    QRect imageRect(20, 20, m_imgWidth/1.3, m_imgHeight/1.3);
    // draw image scaled to widget's size
    painter.drawImage(contentsRect(), m_image, imageRect);
}

#include "main.moc"

int main(int argc, char **argv)
{
    QApplication::setGraphicsSystem("opengl");
    QApplication a(argc, argv);

    Widget w;
    w.resize(800, 400);
    w.show();
    
    return a.exec();
}