= Solution
Let $G=\Lambda$ be the <martini lattice> and $G^*$ its <planar dual graph>. Write $c=p_c^b(G)$ and $c^*=p_c^b(G^*)$. Both are periodic, connected, bounded-degree plane graphs with bounded face sizes. <Dual bond percolation> declares a dual bond open exactly when its crossing primal bond is closed, giving independent dual parameter $1-p$.
First, \b[$c+c^*\le1$]. Suppose instead that the sum exceeds $1$, and choose $1-c^*<p<c$. Both the primal model at $p$ and the dual model at $1-p$ are strictly subcritical. <Sharpness of the percolation transition> gives exponential connection-radius decay on each periodic graph, uniformly over its finitely many vertex types. A crossing of a rectangle of side lengths proportional to $R$ starts at one of $O(R)$ boundary vertices and travels distance of order $R$. A union bound therefore shows that each crossing <probability> tends to zero. But <planar duality for rectangle crossings> says that exactly one of an open primal left-to-right crossing and an open dual top-to-bottom crossing occurs, using the complementary boundary convention. Their <probabilities> sum to $1$. This contradiction proves the first inequality.
For the opposite inequality, we need noncoexistence of primal and dual <infinite percolation clusters>. The four-fold version in Q3(iii) cannot be assumed without change: this decorated <honeycomb lattice> has three-fold rotation symmetry. Here is the necessary <three-fold symmetric primal-dual noncoexistence> argument.
Assume both models percolate at complementary parameters $p,1-p$. For a large disc, the probability of having no exterior infinite arm is at most $\varepsilon$ in each model. Divide its boundary into six arcs by splitting each $120$-degree sector at the same relative angle. In the primal model let $a,b$ be the failure <probabilities> for the two alternating arc types. Rotation symmetry and the <FKG inequality> give
$$
a^3b^3\le\varepsilon,\qquad ab\le\varepsilon^{1/3}.
$$
Slide the splitting point across one sector. Initially the first arc is empty; finally the second is empty. A jump occurs only when an arc gains or loses an edge-crossing point. An arm using that point requires its crossing bond to be open. The <FKG inequality> therefore ensures that a failure <probability> changes by at most a factor $(1-p)^{-1}$ at a single jump: for a newly added arm event $L_x$, $\mathbb P(L_x^c)\ge1-p$ and $\mathbb P(L^c\cap L_x^c)\ge(1-p)\mathbb P(L^c)$. At the first reversal of the order of $a,b$, their ratio is consequently at most $(1-p)^{-2}$. At that splitting point,
$$
\max\{a,b\}\le(1-p)^{-1}\varepsilon^{1/6}.
$$
Thus every one of the six primal arcs has an exterior arm with <probability> tending to $1$. On the same six arcs the dual failure <probabilities> $a^*,b^*$ satisfy $a^*b^*\le\varepsilon^{1/3}$, so at least one dual arc type has failure at most $\varepsilon^{1/6}$; its three rotations have the same bound. Take two of those dual arcs separated by one arc, and the two intervening primal arcs. A union bound gives four alternating primal/dual infinite exterior arms with positive <probability> when $\varepsilon$ is sufficiently small. The separation and finite-closure argument of Q3(iii), together with <percolation uniqueness on an amenable quasi-transitive graph>, gives a contradiction. This proves noncoexistence. The arc balancing is the extra step needed for three-fold symmetry; it is also developed in https://arxiv.org/abs/math/0606149[Bollobás and Riordan's original paper on dual lattices].
If $c+c^*<1$, choose $c<p<1-c^*$. Both complementary models would then percolate, contradicting noncoexistence. Hence $c+c^*\ge1$. Combining the two inequalities proves \b[the dual threshold relation]:
$$
\boxed{p_c^b(\Lambda)+p_c^b(\Lambda^*)=1.}
$$
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