  The k-string tensions are explored in the 4-d $[U(1)]^{N-1}$-invariant dual
Abelian-Higgs-type theory. In the London limit of this theory, the Casimir
scaling is found in the approximation when small-sized closed dual strings are
disregarded. When these strings are treated in the dilute-plasma approximation,
explicit corrections to the Casimir scaling are found. The leading correction
due to the deviation from the London limit is also derived. Its N-ality
dependence turns out to be the same as that of the first non-trivial correction
produced by closed strings. It also turns out that this N-ality dependence
coincides with that of the leading correction to the k-string tension, which
emerges by way of the non-diluteness of the monopole plasma in the 3-d SU(N)
Georgi-Glashow model. Finally, we prove that, in the latter model, Casimir
scaling holds even at monopole densities close to the mean one, provided the
string world sheet is flat.
