Miller et al.'s conjecture on infinite square-free digit walks

For a positive integer $N$, define the right-append map by

$$T_d(N)=10N+d,$$

where $d\in\{0,1,\dots,9\}$. A positive integer is square-free if it is not divisible by any perfect square greater than $1$.

Miller et al.'s conjecture. There exists an infinite sequence of square-free integers

$$\{N_0,N_1,N_2,N_3,\dots\}$$

and digits $d_{k+1}\in\{0,1,\dots,9\}$ such that

$$N_{k+1}=10N_k+d_{k+1}$$

for all $k\geq 0$.

This asks whether one can continue a square-free walk indefinitely by successively appending decimal digits. The question was raised by Miller et al.; despite the result in the paper being previously obtained by Mirsky in 1947, the stated conjecture remains open.