  For tame arbitrary-length toral, also called positive regular, supercuspidal
representations of a simply connected and semisimple $p$-adic group $G$,
constructed as per Adler-Yu, we determine which components of their restriction
to a maximal compact subgroup are types. We give conditions under which there
is a unique such component, and then present a class of examples for which
there is not, disproving the strong version of the conjecture of unicity of
types on maximal compact open subgroups. We restate the unicity conjecture, and
prove it holds for the groups and representations under consideration under a
mild condition on depth.
