  It is known that for a conditional quasi-greedy basis $\mathcal{B}$ in a
Banach space $\mathbb{X}$, the associated sequence
$(k_{m}[\mathcal{B}])_{m=1}^{\infty}$ of its conditionality constants verifies
the estimate $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$ and that if the reverse
inequality $\log m =\mathcal{O}(k_m[\mathcal{B}])$ holds then $\mathbb{X}$ is
non-superreflexive. However, in the existing literature one finds very few
instances of non-superreflexive spaces possessing quasi-greedy basis with
conditionality constants as large as possible. Our goal in this article is to
fill this gap. To that end we enhance and exploit a combination of techniques
developed independently, on the one hand by Garrig\'os and Wojtaszczyk in
[Conditional quasi-greedy bases in Hilbert and Banach spaces, Indiana Univ.
Math. J. 63 (2014), no. 4, 1017-1036] and, on the other hand, by Dilworth et
al. in [On the existence of almost greedy bases in Banach spaces, Studia Math.
159 (2003), no. 1, 67-101], and craft a wealth of new examples of
non-superreflexive classical Banach spaces having quasi-greedy bases
$\mathcal{B}$ with $k_{m}[\mathcal{B}]=\mathcal{O}(\log m)$.
