Counterexample. Let $X=[n]$ where $n\ge8$, and let $$\mathcal S=\{\{1,2\},\ \{1,3\},\ \{2,3\}\}\cup\{\{2,3,x\}:3\lt x\le n\}\subset\mathcal P(X).$$ Then $\mathcal S$ is an intersecting family with no proper shrinking, and $\mathcal S$ is not intersection-efficient, since $$|[\mathcal S]^2_1|=2n-3\lt\binom{n-2}2=|[\mathcal S]^2\setminus[\mathcal S]^2_1|.$$

Counterexample. Let $X=\{1,2,3,\dots,n\}$ where $n\ge8$, and let $$\mathcal S=\{\{1,2\},\ \{1,3\},\ \{2,3\}\}\cup\{\{2,3,x\}:3\lt x\le n\}\subset\mathcal P(X).$$ Then $\mathcal S$ is an intersecting family with no proper shrinking, and $\mathcal S$ is not intersection-efficient, since $$|[\mathcal S]^2_1|=2n-3\lt\binom{n-2}2=|[\mathcal S]^2\setminus[\mathcal S]^2_1|.$$

Another counterexample. Let $X=\{0,1,2,\dots,n\}$ where $n\ge3$, and let $$\mathcal S=\{A\subseteq X:0\in A\}\subset\mathcal P(X).$$ Then $\mathcal S$ is an intersecting family with no proper shrinking, and $\mathcal S$ is not intersection-efficient, since $$|[\mathcal S]^2_1|=\frac{3^n-1}2\lt\binom{2^n}2-\frac{3^n-1}2=|[\mathcal S]^2\setminus[\mathcal S]^2_1|.$$

Counterexample. Let $X=[n]$ where $n\ge8$, and let $$\mathcal S=\{\{1,2\},\ \{1,3\},\ \{2,3\}\}\cup\{\{2,3,x\}:3\lt x\le n\}\subset\mathcal P(X).$$ Then $\mathcal S$ is an intersecting family with no proper shrinking, and $\mathcal S$ is not intersection-efficient, since $$|[\mathcal S]^2_1|=\binom{n-2}2\gt2n-3=\binom n2-\binom{n-2}2=|[\mathcal S]^2\setminus[\mathcal S]^2_1|.$$

Counterexample. Let $X=[n]$ where $n\ge8$, and let $$\mathcal S=\{\{1,2\},\ \{1,3\},\ \{2,3\}\}\cup\{\{2,3,x\}:3\lt x\le n\}\subset\mathcal P(X).$$ Then $\mathcal S$ is an intersecting family with no proper shrinking, and $\mathcal S$ is not intersection-efficient, since $$|[\mathcal S]^2_1|=2n-3\lt\binom{n-2}2=|[\mathcal S]^2\setminus[\mathcal S]^2_1|.$$