Cofinality conjecture for products of boundary Dehn twists

Let $g\geq 2$ and let $b_1,\ldots,b_n$ be curves isotopic to the boundary components of the surface $\Sigma_g^n$. For positive exponents $k_1,\ldots,k_n$, consider the product

$$\prod_{i=1}^n T_{b_i}^{k_i}.$$

Here $T_{b_i}$ denotes the Dehn twist about $b_i$, and an element of a left-ordered group is cofinal if its positive and negative powers are unbounded in the ordering. Cofinality conjecture. Any element of the form

$$\prod_{i=1}^n T_{b_i}^{k_i}$$

for positive exponents $k_1,\ldots,k_n$ is cofinal in every left ordering of $\operatorname{Mod}(\Sigma_g^n)$. This strengthens the preceding result for $\Sigma_g^2$, where the product of the Dehn twists about curves isotopic to the boundary components is cofinal and central in every left ordering.