Effect sizes (eta sqrt) for fixed effects anova table in lmer object - seeking help

Greeting everyone,

Please note I posted this question to SO
https://stats.stackexchange.com/questions/358927/compute-partial-eta2-for-all-fixed-effects-anovas-from-a-lme4-model
but since it's been unanswered and I **honestly** don't know what to do, I
decided to re-post it here.

To make a longer story short (a full reproducible example is posted at SO
and I didn't want to use more space then necessary here):

I used lmer() (actually afex::mixed()) to fit a model for a full factorial
repeated measures experimental data. I'm not interested in random effects
per se - i defined a maximum justified by design structure of random
effects only to control more error variance and my only interest are the
fixed effects - namely Fs from the anova table and corresponding marginal
means.

I've been asked by a reviewer to add effects sizes (etas or omegas) for
each F or t test reported: anova F's from anova(lmer()) and F's and t's
from emmeans calls (pairs and joint_tests) used to explain some higher
order interactions.

I don't feel in a position to argue with the reviewer so I'd like to
provide what she / he wants without any discussion.

My question here is twofold:

   - If there is any citable way to produce eta2 for anovas F tests how can
   I do it? An R package / function / script would be great help.
   - If there is no way computing it - what paper can I cite when answering
   the reviewer why I insisted on skipping effect sizes.

I am aware that my question may sound basic to most of you but i'm really
lost here and would use any detailed help that will allow to produce the
numbers i'm being asked.

If more information is needed please ask here or at SO, time is of the
essence and this group is probably the last place I know of that might help
me.

Thanks for reading this and even a bigger thanks for taking the time to
help me out.
All the best,

Blazej Mrozinski