The infinitude conjecture for lengths of corresponding sequences

Let $T(n)$ denote the minimal length of a corresponding sequence for the non-negative integer $n$.

Infinitude conjecture. For each positive integer $t\neq 2$, the set

$$\{n\in\mathbb{N}_{\geq 0}\mid T(n)=t\}$$

is infinite.

This is presented as a stronger conjecture than the claim that $T(n)$ is unbounded and takes every positive-integer value other than $2$. The paper notes that examples are known for every $t\leq 14$ with $t\neq 2$, and that computation gives $T(n)<15$ for all $n\leq 10000$; the infinitude assertion remains open in the supplied text.