The maximum conjecture for uniform-distribution thresholds

For an integer base $\beta$, let $\Upsilon(\beta)$ be the smallest $K$ such that, for all $k\geq K$, the digits $0,1,\ldots,\beta-1$ are uniformly distributed in the relevant sequence $\left(\Phi_{\beta^k}(n)\right)_{n=1}^{N_k}$.

Maximum conjecture. If coprime bases $\beta_1$ and $\beta_2$ satisfy $\Upsilon(\beta_1)=u$ and $\Upsilon(\beta_2)=v$, then

$$\Upsilon(\beta_1\beta_2)=\operatorname{max}(uv).$$

More generally, for any finite set of pairwise coprime bases $\beta_1,\ldots,\beta_n$ with $\Upsilon(\beta_i)=u_i$, one has

$$\Upsilon(\beta_1,\ldots,\beta_n)=\operatorname{max}(u_1,\ldots,u_n).$$

This is proposed as a generalization of Jacobson's method for combining distribution information in coprime bases. The source gives no proof or resolution, so the conjecture is open.