As an AI language model, I don't have real-time information or updates on travel restrictions and policies. Therefore, I cannot provide you with the most current information about Tibet's tour visit status in 2023.

However, as of my knowledge cutoff in September 2021, Tibet was generally open for tourists but subject to certain restrictions and regulations. To visit Tibet, foreign tourists are required to obtain a Tibet Travel Permit (also known as Tibet Entry Permit) in addition to a Chinese visa. It is typically necessary to join a guided tour organized by a registered travel agency, and individual travel is not permitted. The tour agency will assist in obtaining the necessary permits and handle the required paperwork.

Given that the situation can change over time, it is crucial to stay updated on the latest travel advisories and consult with the Chinese embassy or consulate in your country or a reputable travel agency for the most accurate and up-to-date information on travel regulations and restrictions for Tibet in 2023. They will be able to provide you with the most current guidelines and requirements for planning a visit to Tibet.

Mount Kailash is a popular pilgrimage site located in Tibet, and it attracts a significant number of visitors, especially during the summer months of June and July. Many tourists and religious pilgrims choose this time to undertake the journey due to favorable weather conditions.

During June and July, the weather in the region is relatively mild, with temperatures ranging from 10°C to 20°C (50°F to 68°F) during the day. The mountain paths are usually accessible and not hindered by heavy snowfall, making it easier for travelers to navigate.
If you're interested in undertaking a Mount Kailash tour during June or July, it's advisable to make your travel arrangements well in advance. Contacting reputable travel agencies or tour operators that specialize in Tibet tours would be a good starting point. They can provide you with detailed itineraries, information about permits and visas, transportation options, and accommodation arrangements.
Keep in mind that regulations for travel to Tibet, including Mount Kailash, can change, and it's essential to stay updated on the latest requirements and restrictions imposed by the Chinese government. Additionally, due to the spiritual significance of Mount Kailash for multiple religions, it's important to respect local customs and practices while visiting the area.