This follows from Theorem 1.5 of Alex Smith's paper "The distribution of $\ell^\infty$-Selmer groups in degree $\ell$ twist families I" which states
Suppose $A/\mathbb Q$ is an elliptic curve that fits into eitehr case (1) or (3) of Assumption (1.1). Given any nonicnreasing sequence $r_2 \geq r_4 \geq \dots \geq r_{2^k} \geq \dots$ of nonnegative integers, we have $$ \lim_{H \to\infty} \frac{ \# \{d \in \mathbb Z^{\neq 0} \colon |d|<H \textrm { and } r_{2^K} (A^d) = r_{2^k} \textrm{ for all }k\geq 1\}}{2H} = P^{\textrm{Alt}}(r_2\mid \infty) \cdot \prod_{k=2}^{\infty} P^{\textrm{Alt} }( r_{2^k} \mid r^{2^{k-1}})$$
Choose $r_2 = 2N$ for a large $N$, $r_4 =2N$, $r_8=0$, and thus $r_{2^k}=0$ for all $K>3$. The definition of $P^{\textrm{Alt}}(j\mid n))$ makes it clear that this probability is nonzero as long as $j$ and $n$ have the same parity and $1$ if $j=n=1$, and it can also be checked that $P^{\textrm{Alt}}(j\mid \infty))$ is nonzero for all nonnegative integers $j$, so the product on the right-hand side is zero. Hence the limit on the left-hand side is nonzero. In particular, the numerator on the left-hand side must be nonzero for some $H$, and therefore there exists a $d$ such that the $2$-Selmer rank $r_{2}$ of the quadratic twist $A^d$ is $2N$, the $4$-Selmer rank is $2N$, and the $8$-Selmer rank is $0$.
This implies the algebraic rank is $0$, and so the 4-Sha rank of the Tate-Shafarevich group is at least $2N-2$ (the loss of $2$ coming from the possible $2$-torsion of $A$, though we may choose $A$ to have no $2$-torsion).
