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I'm not sure it's the best way, but you could add the following constraints:$$w_{k,\tau}\le1-w_{\ell,t}\ \forall k,\ell\in\mathcal{L}\ni k\neq\ell;\forall t\in\mathcal{T};\forall\tau\in\left\{ t+1,\dots,t+D_{\ell,k}-1\right\}.$$That's quite a lot of constraints, so I might be inclined to add them lazily (in a loop, solve the LP using a subset of the ...
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These are also known as defined variables (AMPL) or implicit variables (SAS).
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