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You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then the monodromy filtration coincides with the filtration induced by this spectral seq. (There's a nice account of this in Scholl's paper https://www.dpmms.cam.ac.uk/~ajs1005/preprints/weil-preprint1.pdf.) The $E_1$ page of the spectral seq is explicitly given in terms of the components of the special fibre of a semistable model, so you can make explicit what region of the plane they're supported in and you get a bound on the length of the filtration. (I haven't checked whether that bound is the one you want though, the indexing is pretty barbarous.)

Edit. Prompted by Pol's comment I did the computation: the $E_1^{ij}$ terms are supported in a parallelogram with vertices at $(0, 0), (0, d), (2d, 0), (2d, -d)$. So in particular the filtration on $H^n$ has at most $1 + \min(n, 2d-n)$ nonzero graded pieces for any $n$.

You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then the monodromy filtration coincides with the filtration induced by this spectral seq. (There's a nice account of this in Scholl's paper https://www.dpmms.cam.ac.uk/~ajs1005/preprints/weil-preprint1.pdf.) The $E_1$ page of the spectral seq is explicitly given in terms of the components of the special fibre of a semistable model, so you can make explicit what region of the plane they're supported in and you get a bound on the length of the filtration. (I haven't checked whether that bound is the one you want though, the indexing is pretty barbarous.)

Edit. Prompted by Pol's comment, I did the computation: the $E_1^{ij}$ terms are supported in a parallelogram with vertices at $(0, 0), (0, d), (2d, 0), (2d, -d)$. So in particular the filtration on $H^n$ has at most $1 + \min(n, 2d-n)$ nonzero graded pieces for any $n$.

You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then the monodromy filtration coincides with the filtration induced by this spectral seq. (There's a nice account of this in Scholl's paper https://www.dpmms.cam.ac.uk/~ajs1005/preprints/weil-preprint1.pdf.) The $E_1$ page of the spectral seq is explicitly given in terms of the components of the special fibre of a semistable model, so you can make explicit what region of the plane they're supported in and you get a bound on the length of the filtration. (I haven't checked whether that bound is the one you want though, the indexing is pretty barbarous.)

You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then the monodromy filtration coincides with the filtration induced by this spectral seq. (There's a nice account of this in Scholl's paper https://www.dpmms.cam.ac.uk/~ajs1005/preprints/weil-preprint1.pdf.) The $E_1$ page of the spectral seq is explicitly given in terms of the components of the special fibre of a semistable model, so you can make explicit what region of the plane they're supported in and you get a bound on the length of the filtration. (I haven't checked whether that bound is the one you want though, the indexing is pretty barbarous.)

Edit. Prompted by Pol's comment I did the computation: the $E_1^{ij}$ terms are supported in a parallelogram with vertices at $(0, 0), (0, d), (2d, 0), (2d, -d)$. So in particular the filtration on $H^n$ has at most $1 + \min(n, 2d-n)$ nonzero graded pieces for any $n$.

You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then the monodromy filtration coincides with the filtration induced by this spectral seq. (There's a nice account of this in Scholl's paper https://www.dpmms.cam.ac.uk/~ajs1005/preprints/weil-preprint1.pdf.) The $E_1$ page of the spectral seq is explicitly given in terms of the components of the special fibre of a semistable model, so you can make explicit what region of the plane they're supported in and you get a bound on the length of the filtration. (I haven't checked whether that bound is the one you want though, the indexing is pretty barbarous.)

You can bound the filtration length (assuming the WM conj) using the weight spectral sequence of Rapoport--Zink. This is a sp seq converging to $H^*(X_{\overline{k}})$; and if the WM conj holds, then the monodromy filtration coincides with the filtration induced by this spectral seq. (There's a nice account of this in Scholl's paper https://www.dpmms.cam.ac.uk/~ajs1005/preprints/weil-preprint1.pdf.) The $E_1$ page of the spectral seq is explicitly given in terms of the components of the special fibre of a semistable model, so you can make explicit what region of the plane they're supported in and you get a bound on the length of the filtration. (I haven't checked whether that bound is the one you want though, the indexing is pretty barbarous.)