Conjecture on the rarity of block repdigits in Narayana's cows sequence

Let $(N_n)$ be the Narayana sequence, and let an $m$-block repdigit be a positive integer formed by repeating the same decimal block of length $m$ at least twice. Block-repdigit rarity conjecture. For every integer $m\geq2$, the Narayana sequence contains no $m$-block repdigits of length at least $2$. This extends the preceding results for one- and two-block repdigits; whether such repdigits can occur for arbitrary block length remains open.