  The weak geometric P=W conjecture of L. Katzarkov, A. Noll, P. Pandit, and C.
Simpson asserts that for any smooth Betti moduli space $\mathcal{M}_B$ of
complex dimension $d$ over a punctured Riemann surface, the dual boundary
complex $\mathbb{D}\partial\mathcal{M}_B$ is homotopy equivalent to a
$(d-1)$-dimensional sphere. Here, we consider $\mathcal{M}_B$ as a generic
$GL_n(\mathbb{C})$-character variety defined on a Riemann surface of genus $g$,
with local monodromies specified by generic semisimple conjugacy classes at $k$
punctures.
  In this article, we establish the weak geometric P=W conjecture for
$\mathcal{M}_B$ under the condition of being very generic, wherein at least one
conjugacy class additionally satisfies regular semisimplicity.
  Our proof relies on two new results: Firstly, we enhance A. Mellit's cell
decomposition theorem to a stronger form, decomposing $\mathcal{M}_B$ (without
passing to a vector bundle) into locally closed subvarieties of the form
$(\mathbb{C}^{\times})^{d-2b}\times\mathcal{A}$, where $\mathcal{A}$ is stably
isomorphic to $\mathbb{C}^b$. We expect that $\mathcal{A}$ is in general a
counterexample to the Zariski cancellation problem for dimensions $b\geq 3$
over characteristic zero. Secondly, employing a motivic argument, we
demonstrate that the dual boundary complex of any stably affine space (of
positive dimension) is contractible.
  Furthermore, we investigate the treatment of the weak geometric P=W
conjecture for character varieties $\mathcal{M}_B$ that are generic but not
very generic. We observe that this scenario essentially relies on an integral
curious Poincar\'{e} duality (conjecture) for the singular weight cohomology
with compact support of $\mathcal{M}_B$. Additionally, we propose a conjectural
formula by adapting the HLRV conjecture for Voevodsky's motive with compact
support of $\mathcal{M}_B$, supported by illustrative examples.
