The categorical blob versus Soergel conjecture

Let $p$ be a prime, let $l\geq 2$, and fix the data $(e,l,n,\kappa)$ of a generalized blob algebra. For a regular $\boldsymbol{\lambda}\in P_1^l(n)$, let $\mathrm{Blob}^l({\leq \boldsymbol{\lambda}})$ be the category of generalized blob-algebra morphisms between weights in the orbit of $\boldsymbol{\lambda}$ below $\boldsymbol{\lambda}$, and let ${\mathcal{H}}_{BS}^l(\leq \boldsymbol{\lambda})$ be the corresponding finite full subcategory of the characteristic-$p$ diagrammatic Hecke category. Categorical blob versus Soergel conjecture. For each integer $l\geq 2$, there is an equivalence of categories

$$\mathrm{Blob}^l({\leq \boldsymbol{\lambda}}) \cong {\mathcal{H}}_{BS}^l(\leq \boldsymbol{\lambda})$$

sending $\boldsymbol{\mu}$ to $BS(\underline{w}_{\boldsymbol{\mu}})$. This conjecture proposes a categorical realization of the characteristic-$p$ diagrammatic Hecke category through generalized blob algebras; its infinite reformulation is the equivalence in Candidate 4.