The two Igusa zeta function conjectures for degree five

Let $K$ be a local field of residue characteristic not dividing $5!$. Let $q$ be the size of its residue field, set $t=q^{-s}$ and $(a)=1-q^{-a}$, and define

$$I_5(s)=\frac{1}{(1)(2)^2(3)^2(4)^2(5)}\int_{V_5(\mathcal{O}_K)}|\Delta_5(v)|^{s-1}d\mu_{V_5}(v).$$

The degree-five Igusa zeta function conjectures. The following two assertions are conjectured, with (b) stronger than (a): (a) $I_5(s)$ is a power series in $q$ and $t$ with positive integer coefficients independent of $K$, and every nonzero term $q^at^b$ with $b\geq2$ satisfies $a+1<b$; (b) $I_5(s)$ is a power series in $q$ and $t$ with positive integer coefficients independent of $K$, and every nonzero term $q^at^b$ with $b\geq2$ satisfies $2a+1<b$.

These properties would provide the Igusa zeta function input needed to extend the paper's density and smallness results to $d=5$, and the first would imply smoothability results for reduced proper curves admitting degree-five maps to smooth curves. The relevant rationality question remains open in positive characteristic, so the conjectures are not established here.