Let $w_1,\cdots,w_n$ be of elements of $\Omega$, that is the completion of $\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$ for the topology induced by the $-\deg$ valuation.

Assume that $w_1,\cdots,w_n$ are $\mathbb F_q\left(\left(\frac1T\right)\right)$-linearly independent. Can one bound the number of elements of $\mathbb F_q[T] w_1\oplus\mathbb F_q[T] w_2\oplus\cdots\oplus\mathbb F_q[T] w_n$ with degree less than $r$ ($r\in\mathbb R$)?