Conjecture on topological conformal dimensions of fractal rugs

A Jordan arc is a space homeomorphic to $[0,1]$. For a metric space $X$, its topological conformal dimension is denoted by $\dim_{tC}X$.

Fractal-rug conjecture. For any $c\geq 1$ there is a Jordan arc $V_{c-1}$ such that

$$\dim_{tC}(V_{c-1}\times[0,1])=c.$$

This would provide a construction realizing prescribed topological conformal dimensions through products of Jordan arcs with the unit interval. The statement is presented as a conjectural approach, and no resolution is given in the supplied text.