schema markup for multilingual website / subdomains

I just created a translation of my website:

• http://example.com (Original website)
• http://fr.example.com (Translated Version)

I need to know the best way to schema-markup the homepages. On Main Website (English Version), here is the markup I used:

<script type="application/ld+json">
{
"@context": "http://schema.org",
"@type": "Organization",
"@id": "https://website.com/#organization",
"name": "My Website",
"url": "http://example.com/",
"logo": "http://example.com/logo.png",
"contactPoint": {
    "@type": "ContactPoint",
    "telephone": "+2348157437268",
    "contactType": "customer service"
},
"sameAs": [
    "https://www.facebook.com/examplefb",
    "https://twitter.com/exampletwitter",
    "https://instagram.com/exampleig"
]
}
</script>

<script type="application/ld+json">
{
"@context": "http://schema.org",
"@type": "WebSite",
"@id": "http://example.com/#website",
"url": "http://example.com/",
"inLanguage": "en",
"publisher": {"@id": "http://example.com/#organization"}
}
</script>

Now my question is, on the french version homepage, I guess this what I should use:

<script type="application/ld+json">
{
"@context": "http://schema.org",
"@type": "WebSite",
"@id": "http://fr.example.com/#website",
"url": "http://fr.example.com/",
"inLanguage": "fr",
"publisher": {"@id": "https://website.com/#organization"}
}
</script>

But then,

1. should I leave the publisher as "@id": "http://example.com/#organization"? OR
2. should I copy the Organization Script tag into the french homepage also? OR
3. should I just copy the organization details and paste into "publisher" on the french page?

I'm open to corrections if my method/approach is wrong.

Thank You.