Optimal one-symbol redundancy conjecture for Lee-metric function-correcting codes

Let $t\geq 1$ and $q\geq 2t(t+1)+1$. For a function $f$, consider the encoding

$$\operatorname{Enc}(\boldsymbol{u})=(\boldsymbol{u},(pf(\boldsymbol{u}))\bmod q),$$

where $p\in[2t:\lfloor q/2\rfloor]$. The code is required to function-correct for the Lee weight function, the Lee weight distribution function, and the modular sum function. The optimal one-symbol redundancy conjecture. For all such $t$ and $q$, there exists a $p\in[2t:\lfloor q/2\rfloor]$ for which this encoding yields an FCLC for each of the three functions and achieves optimal redundancy $1$. When $q=2t(t+1)+1$, one may take $p=2t+1$. If true, this formalizes the observation that sufficiently large alphabets relative to $t$ permit function-correcting linear codes with a single parity symbol and optimal redundancy. The supplied text gives no resolution.