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Wilkie's proof of o-minimality of exponentiation tells you (without needing to assume Schanuel's Conjecture) tells you that for each $k$ there is a bound $M_k$ such that any exponential polynomial $\sum_{i=1}^k c_ie^{\alpha_i x}$ has at most $M_k$ real zeros [assuming that the $c_i$ and $\alpha_i$ are real.]
Perhaps Khovanskii's work gives an explicit bound on the $M_k$.
Wilkie's proof of o-minimality of exponentiation tells you (without needing to assume Schanuel's Conjecture) tells you that for each $k$ there is a bound $M_k$ such that any exponential polynomial $\sum_{i=1}^k c_ie^{\alpha_i x}$ has at most $M_k$ real zeros [assuming that the $c_i$ and $\alpha_i$ are real.]
Perhaps Khovanskii's work gives an explicit bound on the $M_k$.