Exhaustive classification conjecture for universal diagonal ternary quadratic forms

Let $p$ be a prime and let $\ell$ satisfy $1\leq\ell<p$. A diagonal ternary positive quadratic form is $(p,\ell)$-universal if it represents every natural number congruent to $\ell$ modulo $p$. For $a,b,c>0$, write $\langle a,b,c\rangle$ for the form $ax^2+by^2+cz^2$.

Exhaustive classification conjecture. The complete list of diagonal ternary positive $(p,\ell)$-universal quadratic forms is the one displayed in the source, namely the forms listed for $p=2,3,5,7$, and $101$ with $(p,\ell)=(101,98)$.

The authors say that the computational data suggest that $x^2+2y^2+101z^2$ was the last missing example and that the displayed knowledge is exhaustive. This stronger classification claim is presented as conjectural and is not proved in the supplied text.