  We study well-posedness for the stochastic transport equation with transport
noise, as introduced by Flandoli, Gubinelli and Priola. We consider periodic
solutions in $\rho \in L^{\infty}_{t} L_{x}^{p}$ for divergence-free drifts $u
\in L^{\infty}_{t} W_{x}^{\theta, \tilde{p}}$ for a large class of parameters.
We prove local-in-time pathwise nonuniqueness and compare them to uniqueness
results by Beck, Flandoli, Gubinelli and Maurelli, addressing a conjecture made
by these authors, in the case of bounded-in-time drifts for a large range of
spatial parameters. To this end, we use convex integration techniques to
construct velocity fields $u$ for which several solutions $\rho$ exist in the
classes mentioned above. The main novelty lies in the ability to construct
deterministic drift coefficients, which makes it necessary to consider a convex
integration scheme \textit{with a constraint}, which poses a series of
technical difficulties.
