Unbounded resurgence of fiber products

Let $I$ be a homogeneous polynomial ideal, and let $I^{[k]}$ denote its $k$-fold fiber-product ideal. Write $\rho(I)$ for the resurgence of $I$ and define the asymptotic resurgence by

$$\rho_a(I)=\limsup_{r\to\infty}\frac{\alpha(I^{(r)})}{\alpha(I^r)}.$$

Unbounded fiber-product resurgence conjecture. There exists a homogeneous polynomial ideal $I$ such that

$$\limsup_{k\rightarrow\infty}\rho(I^{[k]})=\infty.$$

The preceding examples show that fiber products can increase resurgence while preserving asymptotic resurgence, and the conjecture asks whether the resurgence of the fiber products of one fixed ideal can become unbounded.