  For each positive integer $Q\in\mathbb{Z}_{\geq 2}$, we prove a multi-valued
$C^{1,\alpha}$ regularity theorem for varifolds in the class $\mathcal{S}_Q$,
i.e., stable codimension one stationary integral $n$-varifolds which have no
classical singularities of vertex density $<Q$, which are sufficiently close to
a stationary integral cone comprised of $2Q+1$ half-hyperplanes (counted with
multiplicity) meeting along a common axis. Such a result furthers the
understanding of the local structure about singularities in the (possibly
branched) varifolds in $\mathcal{S}_Q$ achieved by the author and
N.~Wickramasekera (\cite{minterwick}) and generalises the authors' previous
work in the case $Q=2$ (\cite{minter-5-2}) to arbitrary $Q\in \mathbb{Z}_{\geq
2}$. One notable difference with previous works is that our methods do not need
any a priori size restriction on the (density $Q$) branch set to rule out
density gaps.
