Birget's filling-length characterization of nondeterministic symmetric space

Let $G$) be a finitely generated group and let $f\colon \mathbb{N}\to\mathbb{N}$. A filling length function of a finitely presentable group is the function measuring the maximal length occurring during fillings of null-homotopic words. A nondeterministic symmetric Turing machine is one whose computation relation is symmetric.

Birget's conjecture. There is a nondeterministic symmetric Turing machine accepting the language of words $w$ representing $1$ in $G$ within space

$$f(\ell(w))$$

up to the comparison relation $\preceq$ if and only if $G$ embeds in a finitely presentable group with filling length function $\preceq f$.

This is proposed as an analogue for filling length of the Birget–Ol'shanskii–Rips–Sapir characterization involving polynomial Dehn functions. The affirmative, or “if”, direction is covered by the cited Turing-machine theorem; the converse is the conjectural part.