Thank you for searching excellence. Here is the description of the situation.
No matter the visitor is under page of which language, when an English string (e.g. part of a product name) is entered into the search field, the search result is always displayed. And for the "search result page" itself, visitor can press the language switch button to change the language. However, ...
If the string being entered into the search field is in the 2nd language, the "search reault page" will display result only when the language switch button is switched to the 2nd language. In other words, Search result is found, but being displayed or not is decided by something else.
If the visitor always stick with a single language, this is not an issue; though not ideal.
BTW, I seeked help from both of you and falang on day one. And I received an email from falang yesterday that they are also working on a plugin and will release it soon. so