Cartesianity of exponentiation in infinity-categories

Let $\Space$ be the category of spaces and let $\Cat$ be the category of infinity-categories. For an infinity-category $C$, write $- \to C: \Space \to \Cat$ for the functor sending a space to its corresponding functor category into $C$. A functor is cartesian when it admits cartesian lifts of morphisms. If $f: A \to B$ is a morphism in $\Space$ and $(B,\phi)$ is an object of the relevant total category, its proposed cartesian lift is $(A,\phi \circ f)$.

Cartesianity conjecture. If $C$ is an infinity-category, then

$$- \to C: \Space \to \Cat$$

is cartesian, where the cartesian lift of $f: A \to B$ to $(B,\phi)$ is $(A, \phi \circ f)$.

This conjecture is introduced to support the construction of categories involving mixed-variance applications of the simplicial homotopy principle. The source says that proving it would require showing that cocartesian and cartesian functors are exponentiable; no resolution is given.