This question is about conditions on a mother wavelet that generates a countable familily of child wavelets via scaling and translation, that are both necessary and sufficient for the child wavelets to form a frame in the Hilbert space $L^2(\mathbb{R})$

Here are the precise definitions of these concepts in the context of this question:

Let a mother wavelet be an element $\psi \in L^2({\mathbb{R}})$ with $\|\psi\| = 1$ and a finite admissibility constant $0 \lt C_{\psi} \lt \infty$ that is defined as follows: $$ C_{\psi} = \int_{- \infty}^{\infty} \frac{| \hat{\psi} (\omega) |^2}{|\omega|} d\omega $$ where $\hat{\psi}$ denotes the Fourier transform of $\psi$.

Let $\psi$ be such a mother wavelet and define the countable set of child wavelets $W$ as follows: Let $\sigma \gt 1, \tau \gt 0$ be real numbers and

$$ W := \{ \psi_{j, k}: j, k \in \mathbb{Z}, \psi_{j, k}(t) = \frac{1}{\sigma^{- \frac{j}{2}}} \psi(\frac{t - k \tau \sigma^{-j}}{\sigma^{-j}}) \} $$

Now let a frame in a separable Hilbert space be a countable set of vectors $\{ \phi_j \} $ such that there are constants $a, b \gt 0$ such that for every vector $f$ we have

$$ a \|f\|^2 \le \sum_j | \langle f, \phi_j \rangle |^2 \le b \|f\|^2 $$

My question is: Are there conditions known on the triple $(\psi, \sigma, \tau)$ that are **both necessary and sufficient** for W, the set of child wavelets, to be a frame in $L^2(\mathbb{R})$?

(AFAIK there are conditions that are necessary, and other conditions that are sufficient, known since Ingrid Daubechies published her results in 1990. But there don't seem to be any conditions that are both necessary and sufficient.)