As part of the problem submission to the Epoch AI Open Problems project, David Roe built a verification script that ran various checks on a proposed presentation. These checks were not sufficient to prove the correctness of a presentation, but they provide strong evidence.
Given a marked presentation with four generators $\sigma, \tau, x_0, x_1$, two relations $\tau^\sigma = \tau^2$ and $R(\sigma, \tau, x_0, x_1)$, and the condition that the closed normal subgroup generated by the wild generators $x_0, x_1$ be pro-2, if it is isomorphic to the absolute Galois group of $\mathbb{Q}_2$ then the number of Galois extensions of $\mathbb{Q}_2$ with Galois group $G$ is given by the number of quadruples in $G$ that generate $G$, satisfy the relations, and have $x_0, x_1$ in the 2-core $O_2(G)$ (divided by the size of the automorphism group of $G$). The division is exact: an automorphism fixing such a quadruple fixes a generating set and hence all of $G$, so $\operatorname{Aut}(G)$ acts freely on the quadruples being counted. See Proposition 4.2 of The Inverse Galois Problem for $p$-adic fields.
This provides a strategy: when the number of extensions is known (through the LMFDB database of $p$-adic fields for example), then you can compare the predicted count from the presentation with the known count. If any count disagrees, then the presentation cannot be correct.
In addition to the LMFDB counts, the counts are also known whenever $G$ is a $2$-group by the known presentation for the maximal pro-2 quotient (you can compute the count using the same method for this other presentation, and the two must agree for $2$-groups $G$).
In order to compute the counts, we create a directed system of homomorphisms between finite groups that can possibly arise as Galois groups of $p$-adic fields (they must be potentially $p$-realizable as in Definition 2.1 of loc. cit.). Instead of just counting surjections, we actually store the quadruples (up to automorphism); these can then be lifted along the directed system (with adjustment for the fact that automorphism groups don't map surjectively even when quotienting by a characteristic subgroup).
The test battery used for this release is pinned at padicIGP commit
a8fd266
and mirrored here for audit: gps.txt lists
the 5,402 test groups (sha256 c9232d8dccd3…),
cnts.txt the known extension count for
each (sha256 8fcc008adce2…), and
rel.txt the paper's presentation in the
verifier's input format (sha256 c1b8a0baa875…; the
auxiliary words of Theorem 1.2, with $\omega_2$ replaced by an integer
representative modulo the exponents of the test groups). Full hashes are in
status.json, and the
reproducibility page gives the exact
command for rerunning the comparison.