  We explore Tate-type conjectures over $p$-adic fields. We study a conjecture
of Raskind that predicts the surjectivity of $$ ({\rm NS}(X_{\bar{K}})
\otimes_{\mathbb{Z}}\mathbb{Q}_p)^{G_K}
  \longrightarrow
  H^2_{\rm et}(X_{\bar{K}},\mathbb{Q}_p(1))^{G_K} $$ if $X$ is smooth and
projective over a $p$-adic field $K$ and has totally degenerate reduction.
Sometimes, this is related to $p$-adic uniformisation. For abelian varieties,
Raskind's conjecture is equivalent to the question whether $$
  {\rm Hom}(A,B)\otimes{\mathbb{Q}}_p \,\to\, {\rm Hom}_{G_K}(V_p(A),V_p(B)) $$
is surjective if $A$ and $B$ are abeloid varieties over $K$. Using $p$-adic
Hodge theory and Fontaine's functors, we reformulate both problems into
questions about the interplay of $\mathbb{Q}$- versus $\mathbb{Q}_p$-structures
inside filtered $(\varphi,N)$-modules. Finally, we disprove all of these
conjectures and questions by showing that they can fail for algebraisable
abeloid surfaces.
