  For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION
problem consists in, given a graph $G$ and an integer $k$, decide whether there
exists $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not
contain any of the graphs in ${\cal F}$ as a minor. We are interested in the
parameterized complexity of ${\cal F}$-M-DELETION when the parameter is the
treewidth of $G$, denoted by $tw$. Our objective is to determine, for a fixed
${\cal F}$, the smallest function $f_{{\cal F}}$ such that ${\cal
F}$-M-DELETION can be solved in time $f_{{\cal F}}(tw) \cdot n^{O(1)}$ on
$n$-vertex graphs. We provide lower bounds under the ETH on $f_{{\cal F}}$ for
several collections ${\cal F}$. We first prove that for any ${\cal F}$
containing connected graphs of size at least two, $f_{{\cal F}}(tw)=
2^{\Omega(tw)}$, even if the input graph $G$ is planar. Our main contribution
consists of superexponential lower bounds for a number of collections ${\cal
F}$, inspired by a reduction of Bonnet et al.~[IPEC, 2017]. In particular, we
prove that when ${\cal F}$ contains a single connected graph $H$ that is either
$P_5$ or is not a minor of the banner (that is, the graph consisting of a $C_4$
plus a pendent edge), then $f_{{\cal F}}(tw)= 2^{\Omega(tw \cdot \log tw)}$.
This is the third of a series of articles on this topic, and the results given
here together with other ones allow us, in particular, to provide a tight
dichotomy on the complexity of $\{H\}$-M-DELETION, in terms of $H$, when $H$ is
connected.
