Definite integral of product of modified Bessel functions of the first kind

Question

Assume that $a_1^2+a_2^2=1$ and $b_1^2+b_2^2=1$. We know that $$\int_{\phi=0}^{\pi/2}\sin(\phi)\cos(\phi)\cdot\text{I}_0(z\sin(\phi)a_1)\cdot\text{I}_0(z\cos(\phi)a_2)\mathrm{d}\phi=\frac{2\text{I}_1(z)}{z},$$ where $\text{I}_0(\cdot)$ is the modified Bessel function of the first kind. I was wondering if there exists a similar solution or a proper upper bound for the following argument: $$\int_{\phi=0}^{\pi/2}\sin(\phi)^2\cos(\phi)^2\cdot\text{I}_0(z_1\sin(\phi)a_1)\cdot\text{I}_0(z_1\cos(\phi)a_2)\cdot\text{I}_0(z_2\sin(\phi)b_1)\cdot\text{I}_0(z_2\cos(\phi)b_2)\mathrm{d}\phi$$ A simpler case is when $z_1=z_2$. Thanks.