Boundary-hitting conjecture for Wishart stochastic differential equations

Let $\beta$ and $Q$ be arbitrary $d\times d$ matrices, let $p<\frac{d+1}{2}$, and let $X_t$ be a solution of the Wishart stochastic differential equation with initial condition $X_0=x\in S_d^+$. Write $T_x$ for the first time at which the process hits the boundary of the positive semidefinite cone. Boundary-hitting conjecture. Every such solution hits the boundary in finite time with positive probability:

$$\mathbb P(T_x<\infty)>0.$$

The claim extends a known result for $\beta=0$ and $Q=I$; the source states that the corresponding result for general $\beta$ or $Q\neq 0$ was not known there.