On a conjecture of Levesque and Waldschmidt

One of the first parametrised Thue equations, $$\mid X^3-(n-1)X^{2}Y-(n+2)X{Y}^2-Y^3\mid =1,$$ over the integers was solved by E. Thomas in 1990. If we interpret this as a norm-form equation, we can write this as $$\mid N_{K\mathrm{/}\mathbb{Q}}(X-{\lambda }_{0}Y)\mid =\mid (X-{\lambda }_{0}Y)(X-{\lambda }_{1}Y)(X-{\lambda }_{2}Y)\mid =1$$ if ${\lambda }_0,{\lambda }_1,{\lambda }_2$ are the roots of the defining irreducible polynomial, and $K$ the corresponding number field.\par\medskip Levesque and Waldschmidt twisted this norm-form equation by an exponential parameter $s$ and looked, among other things, at the equation $$\mid N_{K\mathrm{/}\mathbb{Q}}(X-{\lambda }_0^{s}Y)\mid =1.$$ They solved this effectively and conjectured that introducing a second exponential parameter $t$ and looking at $$\mid N_{K\mathrm{/}\mathbb{Q}}(X-{\lambda }_0^s{\lambda }_1^{t}Y)\mid =1$$ does not change the effective solvability. \par\medskip We want to partially confirm this, given that $$\min (\mid 2s-t\mid ,\mid 2t-s\mid ,\mid s+t\mid )>\epsilon \cdot \max (\mid s\mid ,\mid t\mid )>2,$$ i.e. the two exponents do not almost cancel in specific cases.