Problem for update 'locale' props with inertia share with a default local value in HandleInertiaRequest middleware and with SetLocaleController

I want to use a language selector with vuejs3 and inertia.
I don't understand why it doesn't update my current_locale
I have try with inertia link but anything.

HandleInertiaRequests.php

    public function share(Request $request): array
    {
        return array_merge(parent::share($request), [
            ...
            'ziggy' => function () use ($request) {
                return array_merge((new Ziggy)->toArray(), [
                    'location' => $request->url(),
                    'current_locale' => fn () => app()->getLocale(),
                ]);
            },
            'current_locale' => app()->getLocale(),
            'available_locales' => config('app.available_locales'),
            'default_locale' => config('app.locale', 'fr'),
            ...
        ]);
    }

SelectLanguageController.php

    public function setLocale(SetLocaleRequest $request): RedirectResponse
    {
        $locale = $request->validated('locale');

        App::setLocale($locale);
        Session()->put('locale', $locale);
        Inertia::share('current_locale', $locale);

        return redirect()->back();
    }

My languageSelector in vue

<template>
    <span v-for="locale in props.availableLocales"
          :key="`available-locale-${locale}`"
          class="navbar-header__content__languages__item animation text-md "
          :class="props.currentLocale === locale && 'active'"
          @click="setLocale(locale)"
    >
        {{ locale }}
    </span>
</template>

<script setup>
import ...

let props = defineProps({
    availableLocales: {required: true, type: Array},
    currentLocale: {required: true, type: String}
})

let setLocale = (locale) => {
    Inertia.put(route('front.set-locale'), {
            'locale': locale
        },
        {
            onSuccess: () => {
                Inertia.visit(route('front.home'));
            }
        })
}
</script>

web.php

Route::name('front.')->group(function () {

    Route::put('/select-locale', [SelectLanguageController::class, 'setLocale'])
        ->name('set-locale');
});