EF Cannot insert explicit value for identity column in table 'X' when IDENTITY_INSERT is set to OFF

I have to migrate data from one database to another. Keys should be preserved. I'm using EF with an auto-generated .dbmx file. I've read many answers to similar problem and follow those suggestions: I do turn IDENTITY_INSERT ON exactly before inserting the entry; on the database diagram I changed the StoreGeneratedPattern to None on Id column (also tried to do the same with adding corresponding attribute to Id property).

Here is my code:

var myEntity = new MyTable
{
    Id = 12345,
    Name = "Lorem ipsum",
    BrandId = brandId,      
};

MyContext.Database.ExecuteSqlCommand("SET IDENTITY_INSERT MY_TABLE_NAME ON");
MyContext.MyTable.Add(myEntity);
await MyContext.SaveChangesAsync(); //getting error here

When do that in Management Studio, it works just fine. The problem appears only when using Entity Framework.

What is looking weird that I can see in SQL Server Profiler that INSERT_IDENTITY command is executed and insert SQL comes with Id column, but seems is not executed. When I copy paste this SQL query it works just fine.