A Luzin cylinder set gives a tight strongly unbounded coloring

Enumerate $L$ bijectively as $(g_{\beta }{)}_{\beta <\kappa }$ and define $c_{\beta }=g_{\beta }$. For an uncountable $B\subseteq \kappa$, injectivity makes $\{g_{\beta }:\beta \in B\}$ uncountable. Choose its cylinder witness $t$ from F1 and put $n=\mathrm{length}(t)$. For each $m<\omega$, the extension $t^{⌢}(m)$ is a prefix of some $g_{\beta }$ with $\beta \in B$. Hence $\{c_{\beta }(n):\beta \in B,t\subseteq c_{\beta }\}=\omega$, proving strong unboundedness, including when $t$ is empty.

Fix $T\in {\mathcal{T}}_c$ and $B=[T{]}_c$. For each finite string $t$ set $B_t=\{\beta \in B:t\subseteq g_{\beta }\}$. Finite strings form a countable set: their length plus sum of entries stratifies them into finite sets. Remove $N=\bigcup \{B_t:B_t\text{ is countable}\}$ from $B$. By F3, using precisely the countable-choice consequence of A1, $N$ is countable; therefore $B^{\mathrm{\prime }}=B\setminus N$ is uncountable.

Apply F1 to the columns indexed by $B^{\mathrm{\prime }}$ and choose $s$ such that every extension of $s$ occurs among them. Put $T_s=\{t:t\subseteq s\text{ or }s\subseteq t\}$. Every extension of $s$ belongs to $T$, being a prefix of a column in $[T{]}_c$. Every prefix of $s$ also belongs to $T$, by taking one such column extending $s$. Thus $T_s\subseteq T$, without requiring that $T$ itself be prefix-closed.

Some $\beta \in B^{\mathrm{\prime }}$ has $s\subseteq g_{\beta }$. Since $\beta \mathrm{\notin }N$, $B_s$ is uncountable. Every column extending $s$ has all its prefixes comparable with $s$, so $B_s\subseteq [T_s{]}_c$ and $T_s\in {\mathcal{T}}_c$. The fixed family $\{T_s:s\in {\omega }^{<\omega },T_s\in {\mathcal{T}}_c\}$ is countable, and the inclusion and uncountability just proved show it is downward cofinal. This proves tightness with the stated stronger countable bound; injectivity was built into step 1.1.