  Let $\mathscr{k}=\overline{\mathbb{F}_2}$ and let $0\neq\alpha\in
\mathscr{k}$. We present a conjecture supported by computer experimentation
involving the Brenner-Monsky quartic $g_\alpha=\alpha
x^2y^2+z^4+xyz^2+(x^3+y^3)z\in \mathscr{k}[[x,y,z]]$. If true, this conjecture
provides a formula for the Hilbert-Kunz multiplicity and $F$-signature of the
family of four-dimensional hypersurfaces defined by $uv+g_\alpha\in
\mathscr{k}[[x,y,z,u,v]]$ which depends on
$[\mathbb{F}_2(\alpha):\mathbb{F}_2]$, giving an infinite increasing chain of
strict inequalities of $F$-signatures. Additionally, we obtain for any
$t\in\mathbb{N}$ a formula for the Hilbert-Kunz multiplicity and $F$-signature
of the $t$-parameter family of $3t+1$-dimensional hypersurfaces defined by
$uv+\sum\limits_{i=1}^t g_{\alpha_i}(x_i,y_i,z_i)$.
