Hajek in Metamathematics of Fuzzy Logic (1998) states that for every extension of Basic logic a special version of the deduction theorem holds $$A\vDash B\Leftrightarrow\exists n:\vDash\underbrace{A\&\ldots\&A}_{n\text{ times}}\rightarrow B\Leftrightarrow\exists n:\vDash\underbrace{A\rightarrow(\ldots\rightarrow(A}_{n\text{ times}}\rightarrow B)\ldots)$$
In general case (i.e. for any extension of Multiplicative Additive Intuitionistic Linear Logic without $\top$ and $\bot$ — MAILL$^-$), however (cf. Proof Theory for Fuzzy Logics (2008) by Metcalfe, Olivetti, and Gabbay), the following statement holds (using Hajek's notation) $$A\vDash B\Leftrightarrow\exists n:\vDash\underbrace{(A\wedge e)\&\ldots\&(A\wedge e)}_{n\text{ times}}\rightarrow B$$
The question is as follows: does the clause $$A\vDash B\Leftrightarrow\exists n:\vDash\underbrace{A\rightarrow(\ldots\rightarrow(A}_{n\text{ times}}\rightarrow B)\ldots)$$ hold in all MAILL$^-$ extensions?