Finiteness conjecture for universal diagonal ternary quadratic forms

A positive quadratic form is $(k,\ell)$-universal if it represents every natural number congruent to $\ell$ modulo $k$. In particular, let $p$ be prime and let $\ell$ satisfy $1\leq\ell<p$.

Finiteness conjecture. There are only finitely many primes $p$ and integers $\ell$ with $1\leq\ell<p$ for which a diagonal ternary positive $(p,\ell)$-universal quadratic form exists.

The conjecture is motivated by computational evidence: no such forms occur for $11\leq p\leq29$, and among primes up to $1237$ the only remaining case is $(p,\ell)=(101,98)$, where $x^2+2y^2+101z^2$ appears to be universal. The paper presents this as a heuristic conjecture rather than a proved classification.