Non-sliceness conjecture for the exceptional pretzel family

For integers $m\geq 0$, odd integers $a$ and $q_i$ with $a,|q_i|\geq3$ and $a\equiv1,11,37,47,49,59\pmod{60}$, define

$$\mathcal E=\left\{a,-a-2,-\frac{(a+1)^2}{2},q_1,-q_1,\dots,q_m,-q_m\right\}.$$

Let $P(p_1,\dots,p_n)$ be a pretzel knot with one even parameter. Exceptional-family non-sliceness conjecture. If $\{p_1,\dots,p_n\}\subset\mathcal E$, then $P(p_1,\dots,p_n)$ is not slice. This is motivated by substantial evidence, but the exceptional family remains unresolved in the paper.