explicit specifier doesn't seem to work when converting an object to bool

Contextual conversion is special; since C++11, explicit conversion functions will be considered in contextual conversions.

(emphasis mine)

(since C++11)

In the following contexts, the type bool is expected and the implicit conversion is performed if the declaration bool t(e); is well-formed (that is, an explicit conversion function such as explicit T::operator bool() const; is considered). Such expression e is said to be contextually converted to bool.

• the controlling expression of if, while, for;
• the operands of the built-in logical operators !, && and ||;
• the first operand of the conditional operator ?:;
• the predicate in a static_assert declaration;
• the expression in a noexcept specifier;
• the expression in an explicit specifier; (since C++20)
• the predicate of a contract attribute. (since C++20)

That means for if (b2), b2 will be converted to bool implicitly by B::operator bool() even it's declared as explicit.

Read further in your own link. Contextual conversions occur implicitly even for explicit conversions:

Contextual conversions

In the following contexts, the type bool is expected and the implicit conversion is performed if the declaration bool t(e); is well-formed (that is, an explicit conversion function such as explicit T::operator bool() const; is considered). Such expression e is said to be contextually converted to bool.

• the controlling expression of if, while, for;
• the operands of the built-in logical operators !, && and ||;
• the first operand of the conditional operator ?:;
• the predicate in a static_assert declaration;
• the expression in a noexcept specifier;
• the expression in an explicit specifier;
• the predicate of a contract attribute.