The Covering Conjecture for standard $t$-ary expansions

Let $t$ be an integer with $t\geq 2$, and let $n$ and $d$ be positive integers. For an element of ${\mathbb Z}/(t^n-1){\mathbb Z}$, write its standard $t$-ary expansion with digits in the range $0\leq a_i<t$. Say that $b$ covers $a$, written $a\preceq b$, when every digit of $a$ is at most the corresponding digit of $b$; write $a\prec b$ when $b$ strictly covers $a$. Covering Conjecture. If $d$ modulo $t^n-1$ is neither zero nor a power of $t$, then there exist nonzero $a,b\in{\mathbb Z}/(t^n-1){\mathbb Z}$ such that

$$a\prec b\quad\text{and}\quad db\prec da.$$

This elementary number-theoretic conjecture would provide an alternative proof of the universal bound for the relevant Weil-sum valuations. Its general status is open.