Higher-order blow-up differential-form complex conjecture

Let $T$ be a simplex, let $\widetilde{T}$ denote its blow-up, and let $b\mathcal P_1^-\Lambda^k$ be the shadow-form or blow-up Whitney-form complex and $b\mathcal P_r\Lambda^0$ the higher-order blow-up scalar fields. For each flag or face $F$ of $\widetilde{T}$, let $b\mathcal P_r^-\Lambda^k(F)$ be the restriction of the full space to $F$, and let $b\mathring{\mathcal P}_r^-\Lambda^k(F)$ be the subspace of forms vanishing on $\partial F$. Using $\mathring F\cong\prod_j\mathring T_j$, define the corresponding classical spaces $\mathring{\mathcal P}_r^-\Lambda^k(F)$ by the tensor-product construction from the classical FEEC spaces $\mathring{\mathcal P}_r^-\Lambda^k(T_j)$. Higher-order blow-up FEEC conjecture. The spaces $b\mathcal P_1^-\Lambda^k$ and $b\mathcal P_r\Lambda^0$ can be generalized to a differential complex $b\mathcal P_r^-\Lambda^k$ that is exact except at $k=0$. For $r\geq1$, these spaces admit a blow-up geometric decomposition: bases can be constructed whose each $k$-form has a unique minimal face of $\widetilde{T}$ on which it does not vanish. Moreover, for every flag or face $F$ of $\widetilde{T}$, the trace-vanishing spaces $b\mathring{\mathcal P}_r^-\Lambda^k(F)$ are isomorphic to the corresponding classical spaces $\mathring{\mathcal P}_r^-\Lambda^k(F)$. This conjecture proposes an extension of finite element exterior calculus to blow-up simplices; unlike faces of $T$, interiors of faces of $\widetilde{T}$ are products of simplices. The source gives no resolution.