  We investigate the effect of a fixed forbidden clique minor upon the strong
chromatic index, both in multigraphs and in simple graphs.
  We conjecture for each $k\ge 4$ that any $K_k$-minor-free multigraph of
maximum degree $\Delta$ has strong chromatic index at most
$\frac32(k-2)\Delta$. We present a construction certifying that if true the
conjecture is asymptotically sharp as $\Delta\to\infty$. In support of the
conjecture, we show it in the case $k=4$ and prove the statement for strong
clique number in place of strong chromatic index.
  By contrast, we make a basic observation that for $K_k$-minor-free simple
graphs, the problem of strong edge-colouring is "between" Hadwiger's Conjecture
and its fractional relaxation.
  For $k\geq5$, we also show that $K_k$-minor-free multigraphs of edge-diameter
at most $2$ have strong clique number at most $(k-\frac{1}{2})\Delta$.
