Let $A_{\pm}$ be the connection induced by the Levi-Civita connection of $g$ on $\Lambda^{\pm}_{g}$. The Einstein condition implies $A_{+}$ is self-dual and $A_{-}$ anti-self-dual with respect to $g$. Thus

$$|Rm(g)|^{2}dvol_{g}=Tr(F_{A_{+}}\wedge F_{A_{+}}-F_{A_{-}}\wedge F_{A_{-}}).$$

By the tangent cone condition we can easily find a smooth family of spheres $S_{r}$ in $B^{*}$ with the property that as $r$ tends to zero, $(S_{r},r^{-2}g)$ converges smoothly to the round sphere in $\mathbb{R}^{4}$, and the restriction to $(S_{r},r^{-2}g)$ of the connection $A_{\pm}$ converges to the trivial flat connection. Then for any $s<r$

$$\int_{A(s,r)}TrF_{A_{+}}\wedge F_{A_{+}}=CS(A_{+},S_{r})-CS(A_{+},S_{s})(mod\ \mathbb{Z}),$$

where $CS(A,M)=\int_{M}dA\wedge A+\frac{2}{3}A\wedge A\wedge A$ is the Chern-Simons invariant of a connection $A$ over a three manifold $M$, defined modulo $\mathbb{Z}$. By assumption, $CS(A_{+},S_{r})=CS(A_{+},\frac{1}{r}S_{r})\rightarrow 0$ as $r\rightarrow 0$. So we choose $r$ small enough so that for any $s\leq r$ we have $|CS(A_{+},S_{r})|\leq 1/8$ modulo $\mathbb{Z}$. So $\int_{A(s,r)}TrF_{A_{+}}\wedge F_{A_{+}}$ is in $[-1/4,1/4]$ modulo $\mathbb{Z}$, and on the other hand it clearly depends continuously on $s$, so the integral is uniformly bounded for all $s<r$. One can similarly deal with $A_{-}$. Together this implies $\int_{B^{*}}|Rm(g)|^{2}dvol_{g}$ is finite.

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