Reduction of the depth for arithmetic circuits

Although this conjecture is still widely open, it seems more accessible than its Boolean counterpart. The underlying algebraic structure limits the possibilities of computations. In particular, an important result for this model ensures that the low-degree polynomials in VP can be effectively parallelized. Moreover, if we allow a reasonable increase in size, it is also possible to compute them by circuits such that the depth is bounded by a constant.
Then, we will try to see how some lower bounds for these circuits could be achieved by using univariate polynomials. Bürgisser showed that the τ-conjecture which upper-bounds the number of roots of some univariate polynomials, implies lower bounds in arithmetic complexity. But what happens if we try to reduce, as previously, the depth of the considered polynomial? Bounding the number of real roots of certain families of polynomials would allow to separate VP from VNP.