Stabilizing an open book with Anosov piece

It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive stabilizations.

My question goes a bit in the opposite direction:

Let $ \phi:S \to S$ be a diffeomorphism of an oriented compact surface with boundary such that $ \phi|_{\partial S}= id$ and such that $ \phi$ is pseudo-periodic (it does not contain any pseudo-anosov piece in its Nielsen-Thurson decomposition). Suppose that $ \phi$ can be destabilized, that is, there exist a simple closed curve $ c \subset S$ such that $ \phi= t_{c}\circ \tilde{\phi}$ where $ t_c$ is a right-handed Dehn twist around c and $ \tilde{\phi}$ fixes an arc (the co-core of the handle used for stabilizing) whose geometric intersection number with c is $ 1$ .

Can we be sure that $ \tilde{\phi}$ is also pseudo-periodic? Does stabilization preserve pseudo-Anosov-ness somehow?.

What analysis / algorithm helps stabilizing the fit of correlated parameters (but not colinear)?

I have many curves that I want to fit using a convolution of some functions. These functions include Weibull distributions with 2 parameters lambda and k, as well as a function B(t) such as measured curves to fit to model = F(lambda1, k1, kambda2, k2) + B(t)

The main problem here is that even if the lambda’s, k’s and B are not colinear, they can be “kind of” substituted and the optimization can lead to different local minima, with a close final error, but parameters not close at all.

This is a problem because I intend to interpret the value of these parameters as natural characteristics of the objects I observe.

Our actual approach is to minimize the number of parameters, i.e. fixing some of the lambda’s and k’s, as we would do if there were a function linking them. However this is arbitrary + this is a sacrifice as I can not interpret this parameters value anymore.

So question : is there a method / analysis / related problem / science paper dealing with this problem of unstable optimization when parameters are not exactly perpendicular degrees of liberty ?