Existence of a square-filling curve in the topos T

Determine whether the parameterized realizability topos T = PRT(𝕂, M), where 𝕂 is the ppca of oracle-computable partial maps and M is the set of oracles representing a Miller sequence, admits a surjective morphism I → I × I (a square-filling curve), with I denoting the closed unit interval object of Dedekind reals in T.

Background

In Section 6.2 the authors discuss counting the Hilbert cube I^N. A classical route would compose a surjection N → I with a square-filling curve I → I × I (and then iterate) to enumerate I^N. However, they note that constructing such a square-filling curve is not generally possible intuitionistically: there is no such curve in the topos of sheaves on the closed unit square, though countable choice suffices.

For their topos T = PRT(𝕂, M), built from oracle-computable realizers with oracles ranging over representations of a Miller sequence, the authors avoid relying on a space-filling curve and instead prove directly that the Hilbert cube is countable via a realizability flattening argument (Lemma 6.5 and Theorem 6.6). Nevertheless, they explicitly state that it is unknown whether a square-filling curve exists in T.