Generalized pebbling product conjecture for weighted stars

Let $G$ be a graph and $n\ge 1$. Let $k,l\in\mathbb{N}$ satisfy $2\le k\le l$. The pebbling number of a target vertex $t$ is denoted by $\pi(G,t)$, and $\tau_{2,l}(G,t)$ denotes the corresponding two-pebbling parameter with edge weight $l$. Let $R_n^{(k)}$ be the star-like directed graph with $n$ leaves, all edges directed toward its central vertex $s$ and having weight $k$. If

$$\pi(G,t)\le p\qquad\text{and}\qquad\tau_{2,l}(G,t)\le 2p,$$

then the following bounds hold:

Generalized weighted-star pebbling conjecture.

$$\pi\left(R_n^{(k)}\mathbin{\hspace{2pt}\raisebox{1pt}{\scalebox{0.6}{$\square$}}\hspace{2pt}}G,(s,t)\right)\le (nk-n+1)p$$

and

$$\tau_{2,k}\left(R_n^{(k)}\mathbin{\hspace{2pt}\raisebox{1pt}{\scalebox{0.6}{$\square$}}\hspace{2pt}}G,(s,t)\right)\le (nk-n+1)\cdot 2p.$$

The claim generalizes the preceding product results for arrow graphs and hypercubes, and asks whether the weighted-star analogue of the known pebbling-number formula extends to products with an arbitrary graph satisfying the stated pebbling and two-pebbling bounds.