The scalar proportionality conjecture for the qKZB heat kernel

Let $U({\vec{z}},\tau,p)$ be the function defined in Corollary 1, namely

$$U({\vec{z}},\tau,p)=\alpha\otimes Q_{\tau+p}\otimes\alpha\bigl(u({\vec{z}},\tau,\tau+p)\otimes u({\vec{z}},\tau+p,p)\bigr).$$

Here $u$ is the function appearing in the qKZB heat equation, $\alpha$ is multiplication by the function $\alpha(\lambda)$, and $Q_{\tau+p}$ is the operator introduced in the definition of the heat operator. Scalar proportionality conjecture. There is a constant $C$ such that

$$U({\vec{z}},\tau,p)=C u({\vec{z}},\tau,p).$$

This asserts that the composition of the two qKZB heat kernels is proportional to the single heat kernel with combined parameters. The supplied text does not state whether the claim has been proved or disproved.